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July 2022 Hidden symmetries and limit laws in the extreme order statistics of the Laplace random walk
Jim Pitman, Wenpin Tang
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Ann. Probab. 50(4): 1647-1673 (July 2022). DOI: 10.1214/22-AOP1572

Abstract

This paper is concerned with the limit laws of the extreme order statistics derived from a symmetric Laplace walk. We provide two different descriptions of the point process of the limiting extreme order statistics: a branching representation and a squared Bessel representation. These complementary descriptions expose various hidden symmetries in branching processes and Brownian motion which lie behind some striking formulas found by Schehr and Majumdar (Phys. Rev. Lett. 108 (2012) 040601). In particular, the Bessel process of dimension 4=2+2 appears in the descriptions as a path decomposition of Brownian motion at a local minimum and the Ray–Knight description of Brownian local times near the minimum.

Funding Statement

The second author was supported by NSF Grant DMS-2113779 and a start-up grant at Columbia University.

Acknowledgments

We thank G. Schehr for stimulating discussions at the early stage of this work. We also thank one anonymous referee for numerous suggestions which improved the final version of this article.

Citation

Download Citation

Jim Pitman. Wenpin Tang. "Hidden symmetries and limit laws in the extreme order statistics of the Laplace random walk." Ann. Probab. 50 (4) 1647 - 1673, July 2022. https://doi.org/10.1214/22-AOP1572

Information

Received: 1 July 2021; Revised: 1 January 2022; Published: July 2022
First available in Project Euclid: 11 May 2022

Digital Object Identifier: 10.1214/22-AOP1572

Subjects:
Primary: 60G50 , 60J65
Secondary: 60G55 , 60J80

Keywords: branching processes , Brownian embedding , Cox processes , Excursion theory , fluctuation theory , Laplace distribution , limit theorems , order statistics , Path decomposition , Point processes , Random walk , renewal cluster processes , squared Bessel processes

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 4 • July 2022
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